Quasi-stationary distribution for the Langevin process in cylindrical domains, part II: overdamped limit

TL;DR

This paper investigates the overdamped limit of the quasi-stationary distribution for the Langevin process in cylindrical domains, showing that the position marginal converges to the QSD of the overdamped Langevin process.

Contribution

It establishes the convergence of the Langevin process's QSD to the overdamped Langevin process's QSD as friction increases.

Findings

The position marginal of the Langevin QSD converges to the overdamped QSD.

The overdamped limit preserves the quasi-stationary distribution in the position component.

The results extend understanding of Langevin dynamics in bounded domains.

Abstract

Consider the Langevin process, described by a vector (positions and momenta) in $\mathbb{R}^{d}\times\mathbb{R}^d$. Let $\mathcal O$ be a $\mathcal{C}^2$ open bounded and connected set of $\mathbb{R}^d$. Recent works showed the existence of a unique quasi-stationary distribution (QSD) of the Langevin process on the domain $D:=\mathcal{O}\times\mathbb{R}^d$. In this article, we study the overdamped limit of this QSD, i.e. when the friction coefficient goes to infinity. In particular, we show that the marginal law in position of the overdamped limit is the QSD of the overdamped Langevin process on the domain $\mathcal{O}$.